Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 29L (Stochastic Financial Models)

Let $V(n,x)$ be maximal expected utility from dates $n+1,\dots ,N$ plus terminal wealth, given $X_n=x$. Then $V(N,x)=U(x)$ and

With $G=-U=e^{-x}$ and $F=-V$, maximization becomes

Substitution of $F(n+1,x)=B_{n+1}{e}^{-A_{n+1}x}$ separates the infima over investment and consumption. The investment contributes $\gamma ={\inf }_{h}E(e^{-hR})$ after rescaling, and the first-order condition preserves exponential form, with $A_n=A_{n+1}/(1+A_{n+1})$.

Starting from $A_N=1$, recursion gives $A_n=1/(N-n+1)$. The coefficient recursion gives $B_n=(N-n+1){\gamma }^{(N-n)/2}$, including $B_N=1$.

Let $k=N-n+1$. The consumption first-order condition gives